What happens if you try to catch a pokemon when you are at your pokemon inventory limit?

With the new update, whenever your Pokémon inventory is full (i.e. you have 250 Pokémon with you), and you tap on another Pokémon in the wild to enter into the battle arena to capture, it won't enter into battle. Instead, a popup saying "Your Pokémon Box is full. You can buy more space in the shop." will appear followed by an affirmation selection choice saying OK and a secondary choice saying Go To Upgrades.

So, the exact answer to this question is - NO. You won't be able to catch any Pokémon since it escapes the battle arena by showing a popup alert.

About the situation where the Eggs might hatch after you've reached your inventory; I assume that the egg won't hatch unless you've some free space in your bag to carry the Pokémon.

Similar to when you run out of Pokeballs, the game will alert you that you cannot catch any more Pokemon because you have hit the limit to the number of Pokemon you can have. I hit this limit once but forgot to take a screenshot. I will see if I can hit the limit once more and provide one for you.

Based on the other answer by Paras Shah I must have hit the limit with an earlier version of the game.

I finally hit the limit in version 0.33.0, here's a high res screenshot of the alert (click the image for the full resolution):